features overcome the disadvantages of the feature selection and
extraction method for vibration spectrum modeling. However, the
obtained features contain redundancy and complement informa-
tion. Thus, the AGA is applied in LS-SVM to reselect the input sub-
set from the candidate features and the LS-SVM parameters
simultaneously. Thus, the proposed method integrates feature
selection, extraction and modeling. It has been successfully
applied in a laboratory-scale wet ball mill.
The remaining part of the paper is organized as follows.
Section 2 describes the scheme of the proposed feature extraction
and selection method using the vibration frequency spectrum.
Section 3 gives the realization of the proposed method. Section 4
gives the application research results. The conclusions of the
paper are given in Section 5.
2. Scheme of the feature extraction and selection using
vibration frequency spectrum
By using the Fourier transform, the frequency spectrum
becomes a complex. It describes the magnitude and phase of
the mill shell vibration signal. In mill load modeling, phase
information is not important. By discarding the phase informa-
tion, it is possible to simplify the information in a frequency-
domain representation to generate a frequency spectrum or
spectral density. Therefore, based on the analysis of the previous
sections, a feature selection and feature extraction method using
vibration frequency spectrum is proposed to estimate the ML
parameters. The approach consists of local peak features selection
based on MI, spectral segment features extraction based on
clustering and PCA/KPCA, frequency sub-bands selection based
on the MI, and combinational optimization based on AGA.
The strategy diagram is illustrated in Fig. 1.
In Fig. 1, superscript t and f represent the time domain and
frequency domain respectively; subscript v represents shell
vibration; x
t
V
is the original time domain signal; x
f
V
represents
the vibration frequency spectrum; x
f
Vd
represents the spectral
segment, d ¼ 1, ..., D
v
, D
v
is the number of the vibration spectral
segments; z
peak
is the local peaks features; z
i
selpeak
is the local peaks
features for the ith ML parameter selected with MI algorithm; z
sub
is the characteristic frequency sub-bands; z
i
selsub
is the frequency
sub-bands for the ith ML parameter selected with MI algorithm;
z
i
extr
is the extracted features of the spectral segments for the ith
ML parameter with PCA/KPCA algorithm; z
i
¼½z
i
selpeak
, z
i
extr
, z
i
selsub
represents the candidate features for the ith ML parameter;
l
and
g
represent the candidate parameters of the soft sensor
model;
l
i
,
g
i
and z
i
sel
represent the selected models’ parameters
and the input sub-set for the ith ML parameter using AGA based
combinational optimization algorithm; y
i
and
^
y
i
represent the real
and the estimate of the ith ML parameter and i¼1, 2 and 3
represent MVBR, PD and CVR respectively.
Fig. 4 shows that the input and output of the soft sensor model
are x
t
V
and y
i
respectively. The functions of different modules are
shown as follows:
(1) Local peak feature selection based on MI: calculate the features
of the local peaks and select different features using MI-based
feature selection method for different ML parameters.
(2) Spectral segment feature extraction based on clustering and
PCA/KPCA: partition the spectrum into several segments
automatically using the modified spectral segment clustering
algorithm, and extract the linear and nonlinear features of
different spectral segments using PCA/KPCA algorithm.
(3) Frequency sub-band selection based on the MI: select characteristic
frequency sub-bands using MI based feature selection method for
different ML parameters.
(4) Combinational optimization based on AGA: combine these
features as the candidate features and select the input sub-
set and parameters of the soft sensor models simultaneously
using AGA algorithm, which is called combinatorial optimiza-
tion in this paper. There are several reasons to carry out the
combinatorial optimization: (1) the information of ML para-
meters in different features are complementary and redun-
dant; (2) each ML parameter has its own features; (3) for the
same input sub-set, the optimized model parameters can
improve the prediction performance; and (4) the input sub-
set and the model parameters influence each other.
The spectral segmentation is a standard approach used in
many modeling exercises. This paper discusses the vibration
frequency spectrum. The vibration frequency spectrum has the
following properties: the vibration spectrum consists of at least
two spectral segments, one is the vibration mode caused by the
ball mill shell and load inside the mill, another one is caused by
the cyclical impaction of the ball load to mill shell. To deal with
the above properties, some special spectral segmentation meth-
ods are needed. Studies show different spectral segments contain
different ML information. The vibration spectrum consists of lots
of local peaks which can be clustered into several segments.
Each segment has different physical interpretation. A clustering
Shell vibration
Ball mill
Grinding process
Vibration frequecny spectrum
Candidate features
Soft sensor model
Fresher ore
Fresh water
Recycle pulp
Balls
Disturbs
Experimental design
i
y
Adaptive GA
Candidate models’
parameters
f
V
x
peak
z
i
z
Local peaks’
features
Feature
selection
based on MI
selpeak
i
z
Local peak features selection
Sepctral
segments
automatic
partition
Feature
extraction
based on
PCA/KPCA
Spectral segment features extration
Characteristic
frequency
sub-bands
Feature
selection
based on MI
Frequency sub-bands selection
d
f
V
x
...
...
sub
z
selsub
i
z
extr
i
z
i
λ
λ
γ
sel
i
z
i
γ
Combinatorial optimization
based on AGA
t
V
x
...
...
f
V
x
f
V
x
peak
z
i
y
ˆ
Mill load
arameters
Fig. 1. Strategy diagram of the proposed approach.
J. Tang et al. / Control Engineering Practice 20 (2012) 991–1004 993